Early fault early warning and diagnosis method for rolling bearing

Abstract

The invention relates to a rolling bearing early fault early warning and diagnosis method, which comprises the steps of extracting an envelope component from a bearing vibration signal, constructing a time-delay feedback stochastic resonance optimal model by taking an improved signal-to-noise ratio INSR as an optimization objective function, obtaining an output signal, obtaining a first reconstruction signal through CEEMDAN adaptive decomposition and IMF component screening, and obtaining a second reconstruction signal through the CEEMDAN adaptive decomposition and IMF component screening. Calculating the signal-to-noise ratio ISNR of the vibration signal at the fault characteristic frequency, comparing the signal-to-noise ratio ISNR with a preset initial threshold value, judging whether an early warning is given out or not, and if the early warning is given out, processing the vibration signal by using a multi-wavelet adjacent coefficient adaptive threshold value method to obtain a denoised signal; processing the denoised signal by combining a fast spectral kurtosis method and an ensemble empirical mode decomposition method to obtain a second reconstructed signal; and processing by using improved fast spectrum correlation to obtain a corresponding enhanced envelope spectrum, and comparing the enhanced envelope spectrum with a fault characteristic frequency for identification. Compared with the prior art, accurate early warning and diagnosis can be carried out on early weak faults of the rolling bearing.

Description

Technical Field

[0001] The present invention relates to the technical field of rolling bearing fault detection, and particularly to a method for early fault warning and diagnosis of rolling bearings. Background Art

[0002] As an important supporting component, rolling bearings are widely used in various rotating machinery such as wind turbines, internal combustion engines, and aero-engines. Due to the complex and changeable working environment of rolling bearings, even minor faults of them may directly affect the normal operation state of the equipment. Therefore, realizing early warning and diagnosis of weak faults of rolling bearings based on vibration analysis has gradually attracted wide attention from scholars.

[0003] For the fault warning of rolling bearings, it is mainly to construct a series of health index characteristic quantities by performing time-domain analysis and frequency-domain analysis on vibration signals. For time-domain health index characteristic quantities, it mainly includes two parts: dimensional characteristic quantities and dimensionless characteristic quantities. Among them, the dimensional ones include maximum value, minimum value, peak-to-peak value, variance, etc. These indexes are very sensitive to the health state of rolling bearings and can realize the warning of rolling bearings. For frequency-domain health index characteristic quantities, it is mainly to identify the operating state of rolling bearings by extracting and using fault characteristic frequencies. However, focusing on the early warning of rolling bearings, traditional health index characteristic quantities are insensitive to noise, resulting in their inability to detect early faults of bearings. Therefore, the key to realizing early warning of rolling bearing early faults lies in that the health index constructed based on vibration signals has advantages such as good anti-noise performance and enhanced weak fault characteristics, so as to better track the changes in the operating state of the bearing, and then formulate reasonable thresholds by analyzing the change rules of the health index to realize fault warning.

[0004] For the fault diagnosis of rolling bearings, considering that the fault vibration signals exhibit significant non-stationary and non-linear periodic impact characteristics, traditional diagnostic methods, such as wavelet transform, fast spectral kurtosis, fast spectral correlation, etc., mostly reveal the dynamic variation law of signals from both the time domain and the frequency domain, so as to realize fault identification. The wavelet transform (WT) is a signal processing method based on time-frequency window decomposition. It not only inherits the localization advantages of the short-time Fourier transform but also provides a more flexible "time-frequency" window, which is convenient for matching the dynamic impact characteristics of vibration signals and realizing fault diagnosis. The fast spectral kurtosis monitors the impact components in vibration signals based on the kurtosis index, calculates the spectral kurtosis in each frequency band by means of hierarchical band-pass filtering, and characterizes the magnitude of the kurtosis values on each spectral line in the vibration signal to determine the impact components in the signal and their positions in the frequency domain, so as to effectively calculate the center frequency and bandwidth of the corresponding frequency band, and finally identify the bearing fault through band-pass filtering and demodulation analysis. The fast spectral correlation is based on the fact that periodic impact signals have the property of cyclostationarity. By introducing the short-time Fourier transform, the frequency band search range is extended to the full frequency domain. Its essence is to converge and calculate based on spectral correlation, which improves the diagnostic efficiency.

[0005] Focusing on the early warning of incipient weak faults of rolling bearings, for the existing health index characteristic quantities, only when the operating state of the bearing deteriorates significantly will the index values change significantly for early warning. Also, due to the low signal-to-noise ratio characteristics of the early fault vibration signals of rolling bearings, their fault characteristics are weak and are easily affected by external background noise and are difficult to be perceived by traditional health index characteristic quantities. On the other hand, as a supporting component of rotating machinery, rolling bearings are often coupled with the shafting structure and gear structure. When incipient weak faults occur in them, the collected vibration signals exhibit characteristics such as the superposition of various morphological components and the periodic impact components containing fault information being easily submerged. Therefore, using traditional diagnostic methods has the disadvantage of low fault feature identification accuracy. For example, for wavelet transform, when it is directly used for weak fault diagnosis, the influence of noise must be considered. However, the decomposition layer number and threshold selection of wavelet transform determine the noise reduction effect. Due to the lack of a scientific standard at the present stage, the essence of wavelet transform is a Fourier transform with an adjustable window, so it does not have self-adaptability, which affects the diagnostic accuracy; the fast spectral kurtosis is based on the kurtosis index, and this index is easily affected by noise. When it is used for the diagnosis of weak bearing faults, the periodic impact components extracted by the band-pass filter designed with the optimal center frequency and bandwidth it provides contain a large amount of noise, and the diagnostic effect is not good; although the fast spectral correlation can effectively detect the periodic impact components generated during the fault of rolling bearings, when it is applied to weak fault diagnosis, its full-frequency domain analysis feature is instead extremely easily affected by interference components of various different frequencies, and its anti-noise performance is even inferior to that of fast spectral correlation. Summary of the Invention

[0006] The object of the present invention is to overcome the defects existing in the above-mentioned prior art and provide a method for early fault warning and diagnosis of rolling bearings, which can accurately warn and diagnose weak faults in the early stage of rolling bearings.

[0007] The object of the present invention can be achieved by the following technical solutions: A method for early fault warning and diagnosis of rolling bearings, comprising the following steps:

[0008] S1. Obtain the bearing vibration signal, extract the envelope component therefrom, and construct an optimal model of time-delay feedback stochastic resonance with the improved signal-to-noise ratio INSR as the optimization objective function;

[0009] S2. According to the output signal of the optimal model of time-delay feedback stochastic resonance, through CEEMDAN adaptive decomposition and IMF component screening, obtain the first reconstructed signal;

[0010] S3. Calculate the signal-to-noise ratio ISNR of the first reconstructed signal at the fault characteristic frequency, and compare it with a preset initial threshold to determine whether to issue a warning. If it is determined to issue a warning, execute step S4; otherwise, return to step S1;

[0011] S4. For the bearing vibration signal, use the multi-wavelet adjacent coefficient adaptive threshold method for processing to obtain the denoised signal;

[0012] S5. Combine the fast spectral kurtosis method and the ensemble empirical mode decomposition method to process the denoised signal to obtain the second reconstructed signal;

[0013] S6. Use the improved fast spectral correlation to process the second reconstructed signal to obtain the corresponding enhanced envelope spectrum, and compare it with the fault characteristic frequency to identify the fault characteristics.

[0014] Further, the step S1 specifically includes the following steps:

[0015] S11. According to the internal structure of the rolling bearing, pre-calculate the fault characteristic frequencies of the outer ring, inner ring, rolling elements and cage of the bearing;

[0016] S12. Extract the envelope component from the bearing vibration signal and input it into the initial model of time-delay feedback stochastic resonance. With the improved signal-to-noise ratio INSR as the optimization objective function, determine the optimal parameters through parameter traversal, and then obtain the optimal model of time-delay feedback stochastic resonance.

[0017] Further, the fault characteristic frequencies of the outer ring, inner ring, rolling elements and cage of the bearing in the step S11 are specifically:

[0018]

[0019] where z is the number of rolling elements, f r is the rotational frequency, d is the diameter of the rolling element, D m is the raceway diameter, α is the contact angle, f o , f i , f b and f c are the fault characteristic frequencies of the outer ring, inner ring, rolling elements and cage of the bearing, respectively.

[0020] Further, the initial model of time-delay feedback stochastic resonance in step S12 is specifically:

[0021]

[0022]

[0023] The improved signal-to-noise ratio INSR is specifically:

[0024]

[0025] where β is the feedback strength, τ is the lag time, ζ(t) is Gaussian white noise, V(x) is the potential function, X o (n) is the output signal, f n(i) , i = 1, 2,... n represent the fundamental fault characteristic frequency to be detected and its harmonics, P(f n(i) ) is the sum of the effective powers at the fault characteristic frequency and its harmonics, S is the total power in the local frequency band near the fault characteristic frequency and its harmonics, S - P(f n(i) ) represents the power of the background noise in this local frequency band.

[0026] Further, the specific process of step S12 is:

[0027] Set the initial values of the parameters to be optimized in the initial model of time-delay feedback stochastic resonance, including: feedback strength β, lag time τ, and the optimization calculation step sizes h1 and h2 corresponding to the two parameters, the initial signal-to-noise ratio ISNR_Initial;

[0028] Extract the envelope component X envelop (n) from the fault vibration signal X(n) of the rolling bearing and input it into the initial model of time-delay feedback stochastic resonance to obtain the corresponding output signal X o(n), and calculate the corresponding signal-to-noise ratio ISNR_Output. If ISNR_Output > ISNR_Initial, then set ISNR_Output as the new initial value, traverse the parameter range of the two parameters of the lag time τ and the feedback strength β according to the step sizes of β = β + h1 and τ = τ + h2, obtain the system parameters β and τ when the output signal-to-noise ratio is the largest, save the parameters as the optimal parameters, and further obtain the optimal model of time-delay feedback stochastic resonance.

[0029] Further, the specific process of step S2 is as follows:

[0030] According to the output signal X o (n) of the optimal model of time-delay feedback stochastic resonance, adaptively decompose it into multiple different IMF components by CEEMDAN;

[0031] Calculate the improved signal-to-noise ratio of each IMF component at the fault characteristic frequency and the average improved signal-to-noise ratio of all components, select the IMF components greater than the average signal-to-noise ratio, and obtain the first reconstructed signal X or (n).

[0032] Further, step S4 specifically includes the following steps:

[0033] S41. Perform multi-wavelet decomposition on the given original vibration signal X(n), clarify the decomposition layer L of the original vibration signal, and decompose the vibration signal X(n) on the given scaling function and wavelet function to obtain a series of r-dimensional multi-wavelet coefficients {d j (k)}, which represents the k-th coefficient of the j-th layer of the multi-wavelet;

[0034] S42. Perform adaptive threshold denoising on the obtained multi-wavelet coefficients d j (k). First, group the multi-wavelet coefficients on the corresponding scale dimension {b j，k}, and then expand it to the left and right with the central element in the group to form a large group {B j，k};

[0035] Secondly, use the adjacent coefficient shrinkage rule to determine the adaptive threshold coefficient β j，i of the multi-wavelet coefficients on each scale and each dimension, and use it to process each specific wavelet coefficient of the j-th layer and the i-th dimension in each group {b j，k} Finally, calculate the denoised wavelet coefficients:

[0036]

[0037]

[0038] where τ j，i is the contraction factor;

[0039] S43. For the processed multi-wavelet coefficients use the wavelet function and the scaling function to reconstruct the signal, and obtain the denoised signal X reduced noise (n).

[0040] Furthermore, the determination process of the adaptive threshold coefficient β j，i in step S42 includes:

[0041] First, calculate the total energy E j :

[0042]

[0043] where E j，i is the energy of the wavelet coefficients of the j-th layer and the i-th dimension, and M is the length of the wavelet coefficients;

[0044] Second, define a parameter γ j，i :

[0045]

[0046] to clarify the proportion of the energy of each wavelet coefficient of each layer and each dimension in the total energy;

[0047] Third, according to the proportion γ j，i of the energy of each wavelet coefficient of each layer and each dimension in the total energy, calculate the adaptive threshold coefficient β j，i :

[0048] β j，i = λ j，i × γ j，i

[0049]

[0050] where n l is the length of the wavelet coefficients of the l-th layer and the r-th dimension, d l is the corresponding wavelet coefficient, and median(·) is the median function.

[0051] Furthermore, the specific process of step S5 is: use the traditional fast spectral kurtosis method to extract the periodic impulse component X reduced noise (n) and the residual component X periodic_impulse (n) from the denoised signal X residual (n);

[0052] For the residual component X residual(n), using Ensemble Empirical Mode Decomposition (EEMD) and screening the first three Intrinsic Mode Functions (IMFs) with the largest kurtosis based on the kurtosis criterion;

[0053] Finally, superimpose the first three Intrinsic Mode Functions with the largest kurtosis and the periodic shock component X periodic_impulse (n) to obtain the second reconstructed signal X reconstruction_signal (n).

[0054] Furthermore, the step S6 specifically includes the following steps:

[0055] 1) Calculate the short-time Fourier transform X reconstruction_signal (n) of the second reconstructed signal X STFT (i, f k ):

[0056]

[0057] where N w is the window length, R is the moving step, w[n] is the window function, F s is the sampling frequency of the second reconstructed signal X reconstruction_signal (n), f k = kΔf, k = 0,..., N w -1 is the discrete frequency, Δf = F s / N w is the frequency resolution;

[0058] 2) Calculate the complex envelope X reconstruction_signal (n) of the second reconstructed signal X k with f s as the center and Δf as the bandwidth at iR / F w (i, f k ):

[0059]

[0060] 3) Calculate the cyclic spectrum S reconstruction_signal (n) of the second reconstructed signal X X (f, α):

[0061]

[0062] where L represents the length of the second reconstructed signal X reconstruction_signal (n), α is the cyclic frequency, f is the frequency, and the symbol * represents the complex conjugate;

[0063] 4) Assume that the spectral frequency and the cyclic frequency satisfy \(f = f k = k\Delta f\) and \(\alpha = p\Delta f+\delta\), then \(f - \alpha = f k -\alpha\approx f k-p \), and \(\alpha\approx p\Delta f\), obtaining another form of the complex envelope:

[0064]

[0065] 5) Substitute the expressions in steps 2) and 4) into the cyclic spectrum expression in step 3) to obtain its scanned form \(S X (\alpha,f k ,p)\):

[0066]

[0067] 6) Use the scanned form \(S X (\alpha,f k ,p)\) to calculate the Fast Spectral Correlation (FSC) \(S reconstruction_signal (\alpha,f)\) of the second reconstructed signal \(X x Fast (n)\):

[0068]

[0069] In the formula is the kernel function, \(R w (0)=\vert\vert w\vert\vert 2 ;\)

[0070] 7) Use the formula of the Fast Spectral Correlation (FSC) \(S reconstruction_signal (n)\) of the second reconstructed signal \(X x Fas t(\alpha,f)\) to further obtain its Spectral coherence (SCoh) \(\gamma x (\alpha,f)\):

[0071]

[0072] In the formula \(S x (\alpha,f)\) is the estimated value of \(S x Fas t(\alpha,f)\);

[0073] 8) Use the 1 / 3 binary tree structure strategy to set the decomposition level \(Q\) for the frequency band of the spectral coherence \(\gamma x (\alpha,f)\) obtained in step 7), and divide it into a group of narrow bands where \(f g = g\times Fs / (Nw - 1), g = 0, ……, N w -1, α n = α1, α2, α N , α max , f g is an equally - spaced discrete frequency, α n is an equally - spaced discrete cyclic frequency;

[0074] 9) For any spectral frequency f g , let χ χ (n, g), n = 1, …, N be a binary variable, whose value is determined by the local maximum distribution of the g - th cyclic frequency spectral slice :

[0075]

[0076] where l = ±1, ±2, ±3, ±L, and L is a parameter that controls the sparsity of local maxima in the cyclic frequency spectral slice;

[0077] 10) The matrix χ is composed of elements χ Its non - zero elements correspond to the maxima of the cyclic frequency spectral slices on the entire spectral coherence plane. Given that if α n is a frequency related to the rolling bearing fault frequency, then the amplitudes of most cyclic frequency spectral slices at this cyclic frequency α n are relatively larger than the amplitudes near it. This implies that the n - th row of the matrix χ χ contains more non - zero elements. Therefore, the cyclic frequencies corresponding to the rows of the matrix with more non - zero elements are more likely to be caused by the faults of the rolling bearing. To capture this key information, the row vectors of the matrix χ χ are summed to obtain a column vector η = [η(1), η(2), …… η(N)] T , and its element expression is:

[0078]

[0079] 11) Define a function Amp(n), whose expression is Its function can not only quantify the sum of the amplitudes of local maxima in the n - th and solve the sorting problem of two or more equal elements in the vector η. The n - th element of the column vector η quantifies the number of local maxima contained in the n - th cyclic frequency spectral slice . Let satisfy the following conditions:

[0080] a. For any i < j, holds;

[0081] b. If then there exists a k j and ki + 1 such that and and Amp(k i ) > Amp(k i+1 );

[0082] Then the cyclic frequencies for the first D cyclic frequency spectral slices with the most local maxima are represented as:

[0083]

[0084] where is composed of both the fault-related frequency and the interference frequency. For the value of the parameter D, there is a proportional relationship with the maximum cyclic frequency α max . p1 determines the proportion of the fault-related frequency in the entire cyclic frequency. When the parameter p1 is fixed, the number of fault-related frequencies is proportional to the maximum cyclic frequency, ensuring that its number is not affected by the cyclic frequency resolution;

[0085] 12) Integrate the spectral coherence of the i-th (i = 0, 1, 1.6, 2, 2.6, 3,...) narrow width at the l-th level to obtain the corresponding narrowband improved envelope spectrum (IES) IES l，i (α):

[0086]

[0087] 13) Identify the candidate fault frequencies (CFFs) based on the local maximum distribution of all cyclic spectral slices on the entire spectral coherence plane. Use the energy ratio ER l，i of the energy of all CFFs and IES l，i l，i

[0088]

[0089]

[0090] where is an indicator function;

[0091] 14) Since the value of the diagnostic index ER l，i is positively correlated with the significance degree of the periodic shock characteristics of the corresponding sub-band, the IES l，i with the largest value of the diagnostic index ER l，i(α) As the optimal envelope spectrum, it is used to identify the fault characteristic frequency of the rolling bearing.

[0092] Compared with the prior art, the present invention has the following advantages:

[0093] The present invention first constructs a health index that is very sensitive to early faults, and realizes early warning through the processing of fault vibration signals. Specifically, based on the amplitudes at the bearing fault characteristic frequency and its multiples, an improved signal-to-noise ratio (ISNR) is constructed as a health index to measure the operating state of the bearing. It can quantitatively reflect the proportion of the energy of the fault frequency component in the total energy, and uses the time-delayed feedback stochastic resonance (TFSR)-adaptive noise complementary ensemble empirical mode decomposition (CEEMDAN) method to realize early warning of weak faults in rolling bearings; on this basis, a method for extracting weak periodic impact components by wavelet denoising-fast spectral kurtosis-complementary ensemble empirical mode decomposition is given, which can realize noise reduction and extraction of weak periodic impact components in multi-component coupled fault vibration signals; finally, through envelope processing of a specific frequency band of the periodic impact components, noise interference is eliminated, and the weak fault characteristic frequency is enhanced. Thus, by "warning first and then diagnosing" the weak faults of the rolling bearing, accurate warning and diagnosis of the early weak faults of the rolling bearing can be carried out.

[0094] Considering that the fault characteristic frequencies of rolling bearings are mostly concentrated in the low-frequency region compared with high-frequency noise, the present invention designs to utilize the energy transfer characteristic of time-delayed feedback stochastic resonance, and takes the improved signal-to-noise ratio as an index to specifically transfer the high-frequency noise energy to the weak fault characteristic frequency. Specifically, with ISNR as the optimization index, an optimization algorithm is used to perform combined optimization on the system parameters, feedback strength, and lag time of TFSR to obtain the optimal time-delay stochastic resonance system model; the envelope component X envelope (n) of the rolling bearing fault vibration signal X(n) is input into the optimal model, and a signal X o (n) with significantly enhanced energy at the fault characteristic frequency f is output; then the signal X o (n) is processed by CEEMDAN, and the average number of realizations and the screening iteration number are set to obtain a series of intrinsic mode functions (IMFs), and with ISNR as the index, specific IMF components are screened to obtain a first reconstructed signal Xor (n); Calculate the first reconstructed signal X or (n) for its ISNR value. When it exceeds the set threshold, an early warning of bearing faults is issued, which can greatly improve the reliability of early warning.

[0095] The present invention proposes a method for extracting weak periodic impact components by wavelet denoising - fast spectral kurtosis - ensemble empirical mode decomposition for the original rolling bearing fault vibration signal X(n). It first uses multi-wavelet decomposition to obtain the multi-wavelet coefficients of the original signal, and realizes the denoising process of the vibration fault signal by calculating the adaptive threshold of adjacent multi-wavelet coefficients, effectively improving the deficiency that the kurtosis index in the fast spectral kurtosis algorithm is sensitive to impacts but vulnerable to noise interference; then uses fast spectral kurtosis to decompose the denoised signal into a filtered component with the maximum spectral kurtosis and a residual component; for the residual component still reflecting the impact form locally, uses ensemble empirical mode decomposition and extracts several IMF components based on the kurtosis criterion, and reconstructs them with the above-obtained filtered signal, so as to realize the extraction of weak periodic impact components in the multi-component coupled vibration fault signal.

[0096] The present invention also proposes an improved Enhanced Fast Spectral Correlation (EFSC), which is a two-dimensional Fourier transform, showing the power distribution of periodic impact components relative to the spectral frequency f and the cyclic frequency α. Compared with the existing fast spectral correlation based on two-dimensional Fourier transform, by effectively searching for the local maximum amplitude and its corresponding cyclic frequency in all cyclic frequency spectral slices, enhancing the weak modulation phenomenon, and then extracting the specific modulation band containing the most fault information, realizing the accurate positioning of the sub-band containing the fault frequency, improving the recognition accuracy of early bearing fault frequencies, and finally achieving the diagnosis of early weak faults. BRIEF DESCRIPTION OF THE DRAWINGS

[0097] Figure 1 is a schematic diagram of the method flow of the present invention;

[0098] Figure 2 is a schematic diagram of the early warning process in the present invention;

[0099] Figure 3 is a schematic diagram of the diagnosis process in the present invention. DETAILED DESCRIPTION OF THE INVENTION

[0100] The present invention will be described in detail below with reference to the drawings and specific embodiments.

[0101] Embodiment

[0102] It should be noted that this application is proposed based on the National Natural Science Foundation of China project 62303300, "Time-varying non-stationary fault feature extraction method for key components of wind turbine gearboxes under large rotational speed fluctuations". In view of the non-stationary characteristics of the fault vibration signals of rotating machinery, which present multi-component coupling, and the fault information of rolling bearings is contained in periodic impact components. However, the amplitude of early faults is very weak, so it is difficult to directly identify the fault characteristic frequencies from the frequency domain. Therefore, this application proposes a method for early fault warning and diagnosis of rolling bearings, as Figure 1 shown, which includes the following steps:

[0103] S1. Obtain the bearing vibration signal, extract the envelope component therefrom, and construct an optimal model of time-delay feedback stochastic resonance with the improved signal-to-noise ratio INSR as the optimization objective function;

[0104] S2. According to the output signal of the optimal model of time-delay feedback stochastic resonance, through CEEMDAN adaptive decomposition and IMF component screening, obtain the first reconstructed signal;

[0105] S3. Calculate the signal-to-noise ratio ISNR of the first reconstructed signal at the fault characteristic frequency, and compare it with a preset initial threshold to determine whether to issue a warning. If it is determined to issue a warning, execute step S4; otherwise, return to step S1;

[0106] S4. For the bearing vibration signal, use the multi-wavelet adjacent coefficient adaptive threshold method for processing to obtain the denoised signal;

[0107] S5. Combine the fast spectral kurtosis method and the ensemble empirical mode decomposition method to process the denoised signal to obtain the second reconstructed signal;

[0108] S6. Use the improved fast spectral correlation to process the second reconstructed signal to obtain the corresponding enhanced envelope spectrum, and compare it with the fault characteristic frequency to identify the fault characteristics.

[0109] This embodiment applies the above technical solution, as Figure 2 and Figure 3 shown, mainly including a warning process and a diagnosis process. Specifically:

[0110] First, it is necessary to obtain the internal structure of the rolling bearing, acquire the basic parameters in the rolling bearing and some operating parameters of the rotating shaft where the rolling bearing is located, and calculate the corresponding fault characteristic frequencies using the following formula. Among them, z represents the number of rolling elements, f r represents the rotational frequency, d represents the diameter of the rolling element, D m represents the raceway diameter, and α represents the contact angle. f o 、f i 、f b and f cRespectively represent the fault characteristic frequencies of the outer ring, inner ring, rolling elements, and cage of the bearing.

[0111]

[0112] II. For the collected rolling bearing fault vibration signal X(n), where n = 1, 2, 3, ..., N and N is the signal length, through envelope demodulation, its envelope component X envelop (n) is input into the expression (2) of the time-delay feedback stochastic resonance model TFSR. The meanings of each parameter are as follows: β is the feedback strength, τ is the lag time, ζ(t) is Gaussian white noise, and the potential function is X o (n) is the output signal. To achieve the best early fault warning effect for rolling bearings, this scheme proposes to use the improved signal-to-noise ratio INSR as the optimization objective function, as shown in formula (3). The meanings of each parameter are as follows: f n(i) , i = 1, 2, … n represent the fault characteristic fundamental frequencies and their harmonics to be detected, that is, they can be the characteristic frequencies of the outer ring, inner ring, rolling elements, and cage. P(f n(i) ) is the sum of the effective powers at the fault characteristic frequencies and their harmonics. S is the total power in the local frequency band near the fault characteristic frequencies and their harmonics. S - P(f n(i) ) represents the power of the background noise in this local frequency band. Determine the optimal feedback strength, lag time, and other parameters in the model through parameter traversal. The specific process is as follows.

[0113]

[0114]

[0115] 1) Preprocessing of the signal to be analyzed. For the collected rolling bearing fault vibration signal X(n), extract its envelope component X envelop (n) and then input it into the initial time-delay feedback stochastic resonance model. To make the output signal of the system have the best effect, it is necessary to adjust the feedback strength β and lag time τ of TFSR.

[0116] 2) Initial values of the parameters to be optimized in the model. In this scheme, the feedback strength β ∈ (-20000, 20000), the lag time τ ∈ (0, 0.0003), the optimization calculation step sizes h1 and h2 of the two parameters are 5000 and 0.00001 respectively, and the initial signal-to-noise ratio ISNR_Initial = -40dB. Input the above parameters into the TFSR initial model.

[0117] 3) Optimization of the model parameters. Input the envelope component X envelop (n) into the TFSR initial model to obtain the corresponding output signal X o(n), and calculate the corresponding signal-to-noise ratio ISNR_Output;

[0118] If ISNR_Output > ISNR_Initial, then let ISNR_Output be the new initial value, traverse the parameter range of the two parameters of the lag time τ and the feedback strength β according to the step sizes of β = β + h1 and τ = τ + h2, and obtain the system parameters β and τ when the output signal-to-noise ratio in the TFSR model is the largest, and save the parameters.

[0119] 3) Optimal model output. Input the signal into the time-delay feedback model corresponding to the optimal parameters to obtain the final output signal X o (n), whose output signal can effectively improve the signal-to-noise ratio at the fault characteristic frequency, thereby effectively enhancing the energy of the signal component at the weak fault characteristic frequency.

[0120] III. For the signal X o (n) output by the time-delay feedback stochastic resonance model, perform CEEMDAN noise reduction processing. Set the average number of realizations to 500, the screening iteration number to 5000, and the white noise standard deviation to 0.2. Decompose the signal X o (n) adaptively into multiple different IMF components through CEEMDAN. Calculate the improved signal-to-noise ratio of each IMF component at the fault characteristic frequency and the average improved signal-to-noise ratio of all components, and select the IMF components greater than the average signal-to-noise ratio to obtain the first reconstructed signal X or (n).

[0121] IV. Calculate the signal-to-noise ratio ISNR of the first reconstructed signal X or (n) at the fault characteristic frequency. If it is higher than the initial threshold, it can be determined that the bearing is in the early fault stage and an early warning is issued.

[0122] V. After the early warning is issued, use the multi-wavelet adjacent coefficient adaptive threshold method proposed in this scheme to process the original fault vibration signal X(n) of the rolling bearing to obtain the noise-reduced signal X reduced noise (n), and the specific steps are as follows.

[0123] 1) Perform multi-wavelet decomposition on the given original vibration signal X(n). Determine the decomposition level L of the original vibration signal, and decompose the vibration signal X(n) on the given scaling function and wavelet function to obtain a series of r-dimensional multi-wavelet coefficients {d j (k)}, which represents the k-th coefficient of the j-th layer of the multi-wavelet.

[0124] 2) For the obtained multi-wavelet coefficients d j(k) Perform adaptive threshold denoising on multi-wavelet adjacent coefficients. First, group the multi-wavelet coefficients on the corresponding scale dimension into {b j，k} (the number of group members is defined as 1 in this scheme), and then expand to the left and right with the central element in the group to form a large group {B j，k} (the number of expansions to the left and right is 1 each); second, use the adjacent coefficient shrinkage rule to determine the adaptive threshold coefficient β j，i of the multi-wavelet coefficients on each scale and each dimension, and use it to process each specific wavelet coefficient of the j-th layer and the i-th dimension in each group {b j，k} For the adaptive threshold coefficient β j，i , the selection principle of this scheme is as follows: First, use formula (4) to calculate the total energy E j of the wavelet coefficients at each layer, where E j，i is the energy of the wavelet coefficients of the j-th layer and the i-th dimension, and M is the length of the wavelet coefficients. Second, define a parameter γ j，i , and its expression is formula (5), which means that it clarifies the proportion of the energy of the wavelet coefficients of each layer and each dimension in the total energy. Third, according to the proportion γ j，i of the energy of the wavelet coefficients of each layer and each dimension in the total energy, use formulas (6) - (7) to calculate the adaptive threshold coefficient β j，i , where n l is the length of the wavelet coefficients of the l-th layer and the r-th dimension, and d l is the corresponding wavelet coefficient, where median(·) is the median function. Finally, use formula (8) to calculate the denoised wavelet coefficients, where τ j，i is the shrinkage factor, and its calculation formula is formula (9).

[0125]

[0126]

[0127] β j，i = λ j，i × γ j，i (6)

[0128]

[0129]

[0130]

[0131] 3) Signal reconstruction. Use the wavelet function and the scaling function to complete the signal reconstruction of the processed multi-wavelet coefficients to obtain the denoised signal X reduced noise (n).

[0132] VI. Using the traditional fast spectral kurtosis method to extract the periodic impact component X reduced noise (n) and the residual component X periodic_impulse (n) from the signal X residual (n); secondly, for the residual component X residual (n), use the Ensemble Empirical Mode Decomposition (EEMD) and select the first three intrinsic mode functions (IMFs) with the largest kurtosis based on the kurtosis criterion; finally, superimpose the first three intrinsic mode functions with the largest kurtosis on the periodic impact component X periodic_impulse (n) to obtain the second reconstructed signal X reconstruction_signal (n).

[0133] VII. Process the second reconstructed signal X reconstruction_signal (n) using the improved fast spectral correlation proposed in this scheme to obtain the corresponding enhanced envelope spectrum and realize the identification of the fault characteristic frequency. The specific steps are as follows.

[0134] 1) Calculate the short-time Fourier transform X reconstruction_signal (i, f STFT ) of the second reconstructed signal X k (n) using formula (10) as follows. Where N w is the window length, R is the moving step, w[n] is the window function, F s is the sampling frequency of the second reconstructed signal X reconstruction_signal (n), f k =kΔf, k = 0,..., N w -1 is the discrete frequency, and Δf = F s / N w is the frequency resolution.

[0135]

[0136] 2) Calculate the complex envelope X reconstruction_signal (i, f k ) of the second reconstructed signal X s (n) centered at f w with a bandwidth of Δf at iR / F k using formula (11).

[0137]

[0138] 3) Calculate the cyclic spectrum S reconstruction_signal (f, α) of the second reconstructed signal X X (n) using formula (12), where L represents the signal Xreconstruction_signal (n) length, α is the cyclic frequency, f is the frequency, the symbol * denotes the complex conjugate.

[0139]

[0140] 4) Assume that the spectral frequency and the cyclic frequency satisfy f = f k = kΔf and α = pΔf + δ, then f - α = f k - α ≈ f k-p , and α ≈ pΔf, formula (13) is another form of the complex envelope, where p = N w / (2R), N0 is the symmetric center of the window function w[n].

[0141]

[0142] 5) Substitute the expressions in steps 2) and 4) into the cyclic spectrum expression in step 3) to obtain its scanned form S X (α, f k , p), which is formula (14).

[0143]

[0144] 6) Use the scanned form S X (α, f k , p) to calculate the Fast Spectral Correlation (FSC) S reconstruction_signal (α, f) of the second reconstructed signal X x Fast (α, f), as in formula (15). Where is the kernel function, R w (0) = ||w|| 2 .

[0145]

[0146] 7) Use the formula of the Fast Spectral Correlation (FSC) S reconstruction_signal (α, f) of the second reconstructed signal X x Fast (α, f) to further obtain its Spectral coherence (SCoh) γ x (α, f) formula, which is formula (16), where S x (α, f) is the estimate of S x Fast (α, f).

[0147]

[0148] 8) Use the 1 / 3 binary tree structure strategy to set an appropriate decomposition level Q for the frequency band of the spectral coherence γ x (α, f) and divide it into a group of narrow bands where f g = g×F s / (Nw - 1), g = 0, ……, N w -1, α n = α1, α2, α N , α max , f g are equally spaced discrete frequencies, and α n is an equally spaced discrete cyclic frequency. This scheme suggests that the maximum cyclic frequency α max should cover at least three times the fault characteristic frequency.

[0149] 9) For any spectral frequency f g , let χ χ (n, g), n = 1, …, N be a binary variable, and its value is determined by the local maximum distribution of the g-th cyclic frequency spectral slice . The expression is as shown in formula (17), where l = ±1, ±2, ±3, ±L, and L (it is recommended to take 3 - 5) is a parameter that controls the sparsity of the local maxima in the cyclic frequency spectral slice.

[0150]

[0151] 10) The matrix χ is composed of elements χ Its non-zero elements correspond to the maxima of the cyclic frequency spectral slices on the entire spectral coherence plane. Given that if α n is a frequency related to the rolling bearing fault frequency, then the amplitudes of most cyclic frequency spectral slices at this cyclic frequency α n are relatively larger than those near it, which implies that the matrix χ χ contains more non-zero elements in the n-th row. Therefore, the cyclic frequencies corresponding to the rows of the matrix with more non-zero elements are more likely to be caused by the faults of the rolling bearing. To capture this key information, sum the row vectors of the matrix χ χ to obtain a column vector η = [η(1), η(2), …… η(N)] T , and the expression of its elements is as shown in formula (18).

[0152]

[0153] 11) Define a function Amp(n), and the expression is Its function can not only quantify the n-th The sum of the amplitudes of the local maxima, and solve the sorting problem of two or more equal elements in the vector η. The n-th element of the column vector η quantifies the number of local maxima contained in the n-th cyclic frequency spectral slice in, let satisfy the following conditions:

[0154] a. For any i < j, both hold.

[0155] b. If then there exists k j and k i+1 such that and and satisfy Amp(k i ) > Amp(k i+1 ).

[0156] Then the cyclic frequencies of the first D cyclic frequency spectral slices with the most local maxima can be expressed by formula (19), where can be composed of both fault-related frequencies and interference frequencies. The value of the parameter D in this scheme has a proportional relationship with the maximum cyclic frequency α max . p1 determines the proportion of the fault-related frequencies in the entire cyclic frequency. When the parameter p1 is fixed, the number of fault-related frequencies is proportional to the maximum cyclic frequency, ensuring that its number is not affected by the cyclic frequency resolution.

[0157]

[0158] 12) Integrate the spectral coherence of the i-th (i = 0, 1, 1.6, 2, 2.6, 3,...) narrow width at the l-th level to obtain the corresponding narrowband improved envelope spectrum (IES) IES l，i (α) whose expression is as in formula (20).

[0159]

[0160] 13) Identify the candidate fault frequencies (CFFs) according to the local maximum distribution of all cyclic spectral slices on the entire spectral coherence plane, and use the energy ratio ER l，i of the energy of all CFFs and IES l，i , whose expression is as in formula (21), as a diagnostic index to quantify the fault information contained in the corresponding sub-band, where is an indicator function, and its specific expression is formula (22).

[0161]

[0162]

[0163] 14) Since the value of the diagnostic index ER l，i is positively correlated with the significance degree of the periodic impact characteristics of the corresponding sub-band, finally, this scheme selects the IES l，i with the largest ER l，i (α) as the best envelope spectrum to identify the fault characteristic frequency of the rolling bearing.

[0164] In summary, this scheme takes into account that the rolling bearing is an important supporting component of the rotating machinery. When it has early faults, the vibration signal contains a large number of different morphological components such as noise, impact, and modulation, resulting in the impact component containing fault information being relatively weak in energy. Given that the fault vibration signal of the rotating machinery exhibits the non-stationary characteristic of multi-component coupling, the fault information of the rolling bearing is contained in the periodic impact component. However, the early fault determines that its amplitude is very weak, so it is very difficult to directly identify the fault characteristic frequency from the frequency domain. That is to say, as a component of the rotating machinery, the vibration signal collected by the rolling bearing reflects the operating states of different components, so it often exhibits the characteristic of multi-component coupling. The periodic impact component in the vibration fault signal contains bearing fault information, especially the bearing fault characteristic frequency often appears in the form of a modulation frequency. However, when there is an early weak fault, on the one hand, due to the low energy of the periodic impact component, and on the other hand, there is strong vibration propagation attenuation and noise coupling, resulting in the inability to directly demodulate the impact component to identify the fault characteristics.

[0165] Therefore, this scheme proposes a method for realizing the early warning and diagnosis of early weak faults of rolling bearings. Compared with the traditional methods: (1) The fault early warning strategy proposed in this scheme is not based on denoising and purifying the original vibration signal, but on using the energy transfer characteristic of time-delay feedback stochastic resonance, taking the signal-to-noise ratio of the fault characteristic frequency and its harmonics as an index, transferring the noise energy to the fault characteristic frequency and its harmonic components of interest to obtain the enhancement effect of weak fault characteristics, which has a certain engineering value. (2) The early weak fault diagnosis of rolling bearings based on multi-algorithm fusion proposed in this scheme extracts the periodic impact components from the vibration signal as much as possible by integrating the respective advantages of the existing fast spectral kurtosis and fast spectral correlation, avoiding the deficiencies of a single diagnostic algorithm. Through experimental verification, this scheme is more effective than using any one of these methods alone, making the fault characteristics shown in the envelope spectrum more obvious. The "early warning first, then diagnosis" strategy proposed in this scheme can improve the accuracy and reliability of the intelligent operation and maintenance of rolling bearings, and escort the long-term, safe, and efficient operation of rolling bearings.

Claims

1. An early fault warning and diagnosis method for rolling bearings, characterized in that, It includes the following steps: S1. Obtain the bearing vibration signal, extract the envelope component therefrom, and construct an optimal model of time-delay feedback stochastic resonance with the improved signal-to-noise ratio INSR as the optimization objective function. S2. According to the output signal of the optimal model of time-delay feedback stochastic resonance, through CEEMDAN adaptive decomposition and IMF component screening, obtain the first reconstructed signal. S3. Calculate the signal-to-noise ratio ISNR of the first reconstructed signal at the fault characteristic frequency, and compare it with the preset initial threshold to determine whether to issue an early warning. If it is determined to issue an early warning, execute step S4; otherwise, return to step S1. S4. For the bearing vibration signal, use the multi-wavelet adjacent coefficient adaptive threshold method for processing to obtain the denoised signal. S5. Combine the fast spectral kurtosis method and the ensemble empirical mode decomposition method to process the denoised signal to obtain the second reconstructed signal. S6. Use the improved fast spectral correlation to process the second reconstructed signal to obtain the corresponding enhanced envelope spectrum, and compare it with the fault characteristic frequency to identify the fault characteristics.

2. The early fault warning and diagnosis method for a rolling bearing according to claim 1, characterized in that The specific steps of step S1 include the following steps: S11. According to the internal structure of the rolling bearing, pre-calculate the fault characteristic frequencies of the outer ring, inner ring, rolling elements, and cage of the bearing. S12. Extract the envelope component from the bearing vibration signal and input it into the initial model of time-delay feedback stochastic resonance. With the improved signal-to-noise ratio INSR as the optimization objective function, determine the optimal parameters through parameter traversal, and then obtain the optimal model of time-delay feedback stochastic resonance.

3. A method for early fault warning and diagnosis of a rolling bearing according to claim 2, characterized in that The fault characteristic frequencies of the outer ring, inner ring, rolling elements, and cage of the bearing in step S11 are specifically: where z is the number of rolling elements, f r is the rotational frequency, d is the diameter of the rolling element, D m is the raceway diameter, α is the contact angle, f o 、f i 、f b and f c are the fault characteristic frequencies of the outer ring, inner ring, rolling elements and cage of the bearing, respectively.

4. The early fault warning and diagnosis method of a rolling bearing according to claim 3, characterized in that, The initial model of time-delay feedback stochastic resonance in step S12 is specifically: The improved signal-to-noise ratio INSR is specifically: Among them, β is the feedback strength, τ is the lag time, ζ(t) is Gaussian white noise, V(x) is the potential function, and X o (n) is the output signal, f n(i) , where i = 1, 2, … n represent the fundamental frequency of the fault feature to be detected and its harmonics, and P(f n(i) ) is the sum of the effective powers at the fault feature frequency and its harmonics, S is the total power in the local frequency band near the fault feature frequency and its harmonics, and S - P(f n(i) ) represents the power of the background noise in this local frequency band.

5. A method for early fault warning and diagnosis of a rolling bearing according to claim 4, characterized in that The specific process of step S12 is: Set the initial values of the parameters to be optimized in the initial model of time-delay feedback stochastic resonance, including: feedback strength β, lag time τ, and the optimization calculation step sizes h1 and h2 corresponding to the two parameters, and the initial signal-to-noise ratio ISNR_Initial. Extract the envelope component X of the rolling bearing fault vibration signal X(n), envelop (n), and input it into the initial model of time-delay feedback stochastic resonance to obtain the corresponding output signal X o (n), and calculate the corresponding signal-to-noise ratio ISNR_Output. If ISNR_Output > ISNR_Initial, then let ISNR_Output be the new initial value, traverse the parameter ranges of the two parameters of the lag time τ and the feedback strength β according to the step sizes of β = β + h1 and τ = τ + h2, obtain the system parameters β and τ when the output signal-to-noise ratio is the largest, and save the parameters as the optimal parameters, thereby obtaining the optimal model of time-delay feedback stochastic resonance.

6. The early fault warning and diagnosis method for a rolling bearing according to claim 4, characterized in that The specific process of step S2 is: According to the output signal X o (n) of the optimal model of time-delay feedback stochastic resonance, it is adaptively decomposed into multiple different IMF components by CEEMDAN; Calculate the improved signal-to-noise ratio of each IMF component at the fault characteristic frequency and the average improved signal-to-noise ratio of all components, select the IMF components greater than the average signal-to-noise ratio, and obtain the first reconstructed signal X or (n).

7. A method for early fault warning and diagnosis of a rolling bearing according to claim 1, characterized in that The specific steps of step S4 include the following steps: S41. Perform multi-wavelet decomposition on the given original vibration signal X(n), determine the decomposition level L of the original vibration signal, and decompose the vibration signal X(n) on the given scaling function and wavelet function to obtain a series of r-dimensional multi-wavelet coefficients {d j (k)}, which represents the k-th coefficient of the j-th layer of the multi-wavelet; S42. Perform multi-wavelet adjacent coefficient adaptive threshold denoising on the obtained multi-wavelet coefficients d j (k). First, group the multi-wavelet coefficients on the corresponding scale dimension into {b j,k}, and then expand from the central element within the group to the left and right to form a large group {B j,k}; Secondly, the adaptive threshold coefficient β of multi-wavelet coefficients in each dimension at each scale is determined by using the adjacent coefficient shrinkage rule j,i , and it is used to process each specific wavelet coefficient of the j-th layer and the i-th dimension in each group {b j,k} Finally, the denoised wavelet coefficients are calculated: where τ j,i is the contraction factor; S43. For the processed multi-wavelet coefficients Use the wavelet function and the scaling function to complete the reconstruction of the signal, and obtain the denoised signal X reducednoise (n).

8. A method for early fault warning and diagnosis of a rolling bearing according to claim 7, characterized in that, The adaptive threshold coefficient β in the step S42 j,i The determination process includes: First, calculate the total energy E of the wavelet coefficients of each layer j : Among them, E j,i is the energy of the wavelet coefficient at the j-th layer and the i-th dimension, and M is the length of the wavelet coefficient; Secondly, define a parameter γ j,i : It is used to clarify the proportion of the energy of each wavelet coefficient in each layer and each dimension in the total energy. Again, according to the proportion γ of the energy of each wavelet coefficient in each layer and each dimension in the total energy j,i , the adaptive threshold coefficient β is calculated j,i : β j，i = λ j，i × γ j，i where n l is the length of the r-dimensional wavelet coefficients of the l-th layer, d l is the corresponding wavelet coefficient, and median(·) is the median function.

9. A method for early fault warning and diagnosis of a rolling bearing according to claim 7, characterized in that, The specific process of the step S5 is as follows: using the traditional fast spectral kurtosis method to extract the periodic impact component X reduced noise (b) from the denoised signal X periodic_impulse (n) and the residual component X residual (n); For the residual component X residual (n), the ensemble empirical mode decomposition (EEMD) is used, and the first three intrinsic mode functions (IMFs) with the largest kurtosis are selected based on the kurtosis criterion. Finally, the first three intrinsic mode components with the largest kurtosis are superimposed with the periodic impact component X periodic_impulse (n) to obtain the second reconstructed signal X reconstruction_signal (n).

10. A method for early fault warning and diagnosis of a rolling bearing according to claim 9, characterized in that, The specific steps of step S6 include the following steps: 1) Calculate the second reconstructed signal X reconstruction_signal (n) of the short-time Fourier transform X STFT (i, f k ): Among them, N w is the window length, R is the moving step, w[n] is the window function, F s is the sampling frequency of the second reconstructed signal X reconstructiion_signal (n), f k = kΔf, k = 0, …, N w -1 is the discrete frequency, Δf = F s / N w is the frequency resolution; 2) Calculate the second reconstructed signal X reconstruction_signal (n) centered at f k with a bandwidth of Δf, in the complex envelope X s of iR / F w (i, f k ): 3) Calculate the second reconstructed signal X reconstruction_signal (n) of the cyclic spectrum S X (f, α): where L represents the length of the second reconstructed signal X reconstruction_signal (n), α is the cyclic frequency, f is the frequency, and the symbol * denotes the complex conjugate; 4) Assume that the spectral frequency and the cyclic frequency satisfy \(f = f k = k\Delta f\) and \(\alpha = p\Delta f+\delta\), then \(f - \alpha = f k -\alpha\approx f k-p , and \(\alpha\approx p\Delta f\), obtaining another form of the complex envelope: 5) Substitute the expressions in steps 2) and 4) into the cyclic spectrum expression in step 3) to obtain its scanning form S X (α, f k , p): 6) Use the scanning form S of the cyclic spectrum X (α, f k , p) to calculate the fast spectral correlation S reconstruction_signal (n) of the second reconstructed signal X x Fast (α, f): where is the kernel function, R w (0) = ‖w‖ 2 ; 7) Using the second reconstruction signal X reconstruction_signal (n) of the fast spectral correlation S x Fast (α,f) formula, and further calculate its spectral coherence γ x (α,f): where S x (α, f) is the estimated value of S x Fast (α, f); 8) Use the 1 / 3 binary tree structure strategy to decompose the spectral coherence γ x (α,f) obtained in step 7) into a set of narrow bands by setting the decomposition level Q of the frequency band where f g = g×F s / (N w -1), g = 0, ……, N w -1, α n = α1, α2, α N , α max , f g is equally spaced discrete frequency, α n is equally spaced discrete cyclic frequency; 9) For any spectral frequency f g , let χ χ (n, g), n = 1, …, N be binary variables, whose values are determined by the local maximum distribution of the g-th cyclic frequency spectral slice : Where l = ±1, ±2, ±3, ±L, and L is a parameter that controls the sparsity of local maxima in the cyclic frequency spectrum. 10) A matrix composed of elements χ χ (n,g) whose non - zero elements correspond to the maximum values of the cyclic frequency spectral slices on the entire spectral coherence plane. Given that if α n is a frequency related to the fault frequency of a rolling bearing, then the amplitudes of most cyclic frequency spectral slices at this cyclic frequency α n are relatively larger than those in its vicinity, which implies that the matrix χ χ contains more non - zero elements in the n - th row. Therefore, the cyclic frequencies corresponding to the matrix rows with more non - zero elements are more likely to be caused by the faults of rolling bearings. To capture this key information, the row vectors of the matrix χ χ are summed to obtain a column vector η = [η(1), η(2), …… η(N)] T , and its element expression is as follows: 11) Define a function Amp(n) with the expression Its function is not only to quantify the sum of the amplitudes of the local maxima in the nth and solve the sorting problem of two or more equal elements in the vector η. The nth element of the column vector η quantifies the number of local maxima contained in the nth cycle frequency spectral slice Let Satisfy the following conditions: a. For any i < j, it holds; b. If Then there exists k j and k i+1 such that and and satisfy Amp(k i ) > Amp(k i+1 ); Then the cyclic frequencies of the first D cyclic frequency spectrum slices with the most local maxima are expressed as: In the formula It is composed of the fault-related frequency and the interference frequency at the same time. For the value of parameter D, it has a proportional relationship with the maximum cycle frequency α max There is a proportional relationship. p1 determines the proportion of the fault-related frequency in the entire cycle frequency. When the parameter p1 is fixed, the number of fault-related frequencies is proportional to the maximum cycle frequency, ensuring that its number is not affected by the cycle frequency resolution; 12) Integrate the spectral coherence of the i-th (i = 0, 1, 1.6, 2, 2.6, 3, …) narrow width at the l-th level to obtain the enhanced envelope spectrum IES of the corresponding narrowband l,i (α): 13) Identify candidate fault feature frequencies CFFs according to the local maximum distribution of all cyclic spectral slices on the entire spectral coherence plane, and use the energy ratio ER l,i between the energy of all CFFs and the energy of IES l,i as a diagnostic index to quantify the fault information contained in the corresponding sub-band: wherein is an indicator function; 14) Since the value of the diagnostic index ER l,i is positively correlated with the significance level of the periodic impact characteristics of the corresponding sub-band, the diagnostic index ER l,i value of the largest IES l,i (α) is used as the best envelope spectrum to identify the fault characteristic frequency of the rolling bearing.

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